Dimensionality reduction is required in modern machine learning to condense high-dimensional datasets into lower-dimensional representations while retaining all information. This simplifies data presentation, increases processing efficiency, and unveils patterns or structures that were not visible in the first place. As a non-linear dimensionality reduction method, Isomap stands out. Isomap uses graph theory and classical multidimensional scaling (MDS) to find low-dimensional structures in high-dimensional data. Isomap works best on non-linear manifolds, different from PCA.

In this article, we explore Isomap in-depth, explaining its core principles, how it works, its advantages and limitations, and its practical applications in machine learning.

What is Isomap?

​A non-linear dimensionality reduction method used in data analysis and machine learning is called isomap (isometric mapping). Its purpose is to transfer data from a high-dimensional space to a lower-dimensional one while maintaining the data’s inherent geometric structure. When datasets contain data points that sit on a non-linear manifold—that is, when the relationships between the data points cannot be captured by straightforward linear projections (as with techniques like PCA)—Isomap is very helpful.

Isomap in Machine Learning

Data is not linearly distributed in many real-world applications. Non-linear structures are common in photos, sounds, and time-series data. Complex datasets are problematic for linear methods like PCA, which presume data lie on a straight line or hyperplane. Isomap and other non-linear dimensionality reduction algorithms preserve the data’s intrinsic geometry, which linear methods struggle to represent.

Manifold learning, a non-linear dimensionality reduction technique, assumes high-dimensional data often lie on a lower-dimensional manifold. Manifolds are topological spaces that may have larger global structures than Euclidean space. Different sources of data may be regulated by hidden variables. Isomap is designed to discover lower-dimensional structures and format the data for easier interpretation.

Core Principles of Isomap

The graph-based technique isomap learns a low-dimensional data representation while retaining geodesic distances. Breaking down the main ideas helps:

• Geodesic Distance: For two points in Euclidean space, the straight-line distance is the geodesic distance. The shortest path between two points on a manifold may be a curved path on the manifold. The geodesic distance is the shortest path. In high-dimensional spaces, geodesic distances are hard to compute directly.
• Neighborhood Graph: Each data point is connected to its nearest neighbors in Isomap’s neighborhood graph to approximate geodesic distances. By determining the shortest path through this graph, the geodesic distance may be calculated from the Euclidean distances between points. The graph will connect nearby locations in high-dimensional space but not distant ones.
• Graph-based Shortest Path: Graph-based Once the graph is formed, Isomap calculates the shortest path between all pairs of points. A graph traversal method like Dijkstra’s or Floyd-Warshall can find the shortest path between two points. These algorithms help Isomap estimate geodesic distances.
• Multidimensional Scaling (MDS): Isomap performs classical MDS after obtaining pairwise geodesic distances. Data is represented in a lower-dimensional space using MDS to preserve point distances. Geodesic distances are used by Isomap to embed data in a lower-dimensional space using traditional MDS.

In summary, Isomap approximates manifold geodesic distances using neighborhood graphs and graph-based shortest path techniques. Projecting the data onto a lower-dimensional space via MDS preserves these distances.

Steps Involved in the Isomap Algorithm

The steps of the Isomap algorithm are:

• Neighborhood Graph Construction: Define Euclidean distance-based nearest neighbors for each data point. Build a graph with nodes representing data points and edges the nearest neighbors’ Euclidean distances. Each point is usually connected to its k-nearest neighbors or all points within a radius.
• Compute Geodesic Distances: Use a graph-based shortest path algorithm, such as Dijkstra’s, to compute the geodesic distance between each pair of points. The geodesic distance of the manifold can be approximated by finding the shortest path between two locations on the graph.
• Apply Multidimensional Scaling (MDS): Apply classical Multidimensional Scaling (MDS) to obtain a lower-dimensional data representation after computing the geodesic distance matrix. Place the points in a lower-dimensional space so their pairwise distances are near to geodesic distances.
• Output the Low-dimensional Representation: The outcome is a collection of coordinates in 2D or 3D that preserves the data’s local and global structure.

Advantages of Isomap

In contrast to PCA, isomap has various advantages:

• Preserves Global Structure: PCA only records linear correlations, but Isomap preserves the manifold’s global shape. Isomap preserves geodesic distances to preserve linkages between distant points in high-dimensional space during low-dimensional representation.
• Non-linear Data: For non-linear data on a manifold, isomap is beneficial. In image processing, Isomap can capture the non-linear relationship between pictures of objects from different perspectives on a manifold.
• Versatile: Image, voice, genomics, and time-series data can be used with isomap. Effective when data naturally lies on a curved, non-linear manifold rather than a flat plane.

Disadvantages of Isomap

The powerful Isomap approach has its drawbacks:

• Complexity: Isomap is computationally expensive, especially for huge datasets. Large datasets demand significant computational resources to build the neighborhood graph and compute the shortest paths between all pairs of points. For huge datasets, Isomap is unsuitable since shortest path techniques take quadratic or worse time.
• Sensitivity to Parameters: The number of neighbors utilized to generate the neighborhood graph affects Isomap’s performance. Few neighbors may not capture the manifold’s structure, while too many may over-smooth. Noise in the data might also impair algorithm performance.
• Dependence on Local Linear Approximation: Isomap believes the manifold is locally approximated by Euclidean geometry, so each local neighborhood is flat. While this is true for many datasets, Isomap may fail if the data is substantially non-linear or has significant curvature.

Applications of Isomap

Isomap is used in several machine learning fields despite its limitations:

• Image Processing: Computer vision uses isomap for face recognition, image reconstruction, and object identification. Face photos taken from different perspectives or lighting circumstances typically sit on a low-dimensional manifold, which Isomap can find.
• Speech and Audio Processing: Isomap can minimize the dimensionality of the feature space while keeping speech patterns in speech recognition due to the non-linear nature of audio signals.
• Genomics: Isomap analyzes non-linear gene expression data in bioinformatics. Isomap can find significant patterns like gene expression profile clusters by lowering data dimensionality.
• Time-series Data: While keeping temporal structure, isomap has also been applied to time-series data to minimize dimensionality of trajectories.

Conclusion

Machine learning often uses isomap to reduce the dimensionality of complex, high-dimensional data on non-linear manifolds. Using geodesic distances to preserve the data’s fundamental geometry, Isomap can find useful low-dimensional representations that reveal its structure. Due to its computational cost and parameter sensitivity, it is not ideal for large datasets or circumstances where the local linearity assumption does not apply. Although limited, Isomap is a useful machine learning tool for computer vision, speech processing, and bioinformatics.